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Exploration of Special Matrices

Introduces five series of matrices (with MATLAB code to generate them) and asks students to find out everything they can about them (determinant, eigenvalues, eigenvectors) and prove the results. Solutions available to instructors on request.

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This one-page document introduces five general \(n\times n\) matrices, and asks students to find general formulas for the determinant, eigenvalues, and eigenvectors for each. The matrices are the "checkerboard" (alternating zeroes and ones), "cyclic" (ones on diagonal above main diagonal and in lower left, zeroes elsewhere), "letter N" (ones in the left and right columns and on main diagonal), "letter X" (ones on the main diagonal and antidiagonal, zeroes elsewhere), and "Hankel consecutive integers" (row \(j\) has entries \(j\), \(j+1\), ..., \(j+n-1\)). MATLAB code to produce each matrix is included. Students are asked to generate examples, form conjectures for the general form of the determinant, eigenvalues, and eigenvectors, and then prove their conjectures. Professors may request solutions from the author.